Neighborhood Complexes of Kneser Graphs, $KG_{3,k}$

TL;DR

This paper investigates the topological structure of neighborhood complexes of Kneser graphs $KG_{3,k}$, showing they are homotopy equivalent to a wedge of spheres, and constructs a maximal subgraph with a related deformation retraction.

Contribution

It proves the homotopy type of neighborhood complexes of $KG_{3,k}$ and constructs a maximal subgraph with a deformation retraction onto a simpler neighborhood complex.

Findings

Neighborhood complex of $KG_{3,k}$ is homotopy equivalent to a wedge of spheres.

Number of spheres in the wedge is explicitly calculated.

Constructs a maximal subgraph with a deformation retraction onto a simpler complex.

Abstract

In this article, we prove that the neighborhood complex of the Kneser graph $KG_{3,k}$ is of the same homotopy type as that of a wedge of $\frac{(k+1)(k+3)(k+4)(k+6)}{4}+1$ spheres of dimension $k$. We construct a maximal subgraph $S_{3,k}$ of $KG_{3,k}$, whose neighborhood complex deformation retracts onto the neighborhood complex of $SG_{3,k}$.